Since, heat flow from hotter to colder bodies is irreversible, reversible processes must involve heat flow between bodies of the same temperature. It follows that any entropy change of the system must be exactly balanced by that of the heat bath which provided the heat.
Thus, the entropy change of the universe during reversible processes is zero.
During an adiabatic process no heat flows. Thus, we see that the entropy change of a system during for a reversible, adiabatic process is zero.
But note that both qualifiers are needed; the entropy of a non-isolated system change during a reversible process (and the entropy change of the surrounding will compensate), and an irreversible change to an isolated system will increase the entropy.